1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 7.2 Addition and Subtraction with Rational Expressions

2. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Objectives o Add rational expressions. o Subtract rational expressions.

3. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Addition with Rational Expressions For polynomials P, Q, and R, with Q ≠ 0,

4. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Adding Rational Expressions with a Common Denominator Find each sum and reduce if possible. (Note the importance of the factoring techniques we studied in Chapter 6.) Solution

5. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Adding Rational Expressions with a Common Denominator (cont.) Remember, if we use the entire expression in the numerator (or denominator) to reduce, we are left with a factor of 1. Solution

6. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Addition with Rational Expressions To Find the LCM for a Set of Polynomials 1.Completely factor each polynomial (including prime factors for numerical factors). 2.Form the product of all factors that appear, using each factor the most number of times it appears in any one polynomial.

7. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Addition with Rational Expressions Procedure for Adding Rational Expressions with Different Denominators 1.Find the LCD (the LCM of the denominators). 2.Rewrite each fraction in an equivalent form with the LCD as the denominator. 3.Add the numerators and keep the common denominator. 4.Reduce if possible.

8. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Adding Rational Expressions with Different Denominators Find each sum and reduce if possible. Assume that no denominator has a value of 0. Solution In this case, neither denominator can be factored so the LCD is the product of these factors. That is, LCD = (y – 3)(y + 4).

9. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Adding Rational Expressions with Different Denominators (cont.) Now, using the fundamental principle, we have The numerator is not factorable and the expression is reduced.

10. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Adding Rational Expressions with Different Denominators (cont.) Solution First, find the LCD. Step 1: Factor each expression completely.

11. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Adding Rational Expressions with Different Denominators (cont.) Step 2: Form the product of 2, (x + 3) 2 and (x − 3). That is, use each factor the most number of times it appears in any one factorization. Now use the LCD and add as follows.

12. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Adding Rational Expressions with Different Denominators (cont.)

13. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Addition with Rational Expressions Notes Important Note about the Form of Answers In Examples 2a and 2b each denominator is left in factored form as a convenience for possibly reducing or adding to some other expression later. You may choose to multiply out these factors. Either form is correct. For consistency, denominators are left in factored form in the answers in the back of the text.

14. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Subtraction with Rational Expressions Placement of Negative Signs If P and Q are polynomials and Q ≠ 0, then

15. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Subtraction with Rational Expressions For polynomials P, Q, and R, with Q ≠ 0,

16. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Subtracting Rational Expressions with a Common Denominator Find each difference and reduce if possible. Assume that no denominator has a value of 0. Solution Subtract the entire numerator.

17. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Subtracting Rational Expressions with a Common Denominator (cont.) Solution Subtract the entire numerator. Factor and reduce.

18. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Subtracting Rational Expressions with a Common Denominator (cont.) Solution Each denominator is the opposite of the other. Multiply both the numerator and denominator of the second fraction by  1 so that both denominators will be the same, in this case x  5.

19. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Subtracting Rational Expressions with a Common Denominator (cont.)

20. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Subtraction with Rational Expressions Notes COMMON ERROR Many beginning students make a mistake when subtracting rational expressions by not subtracting the entire numerator. They make a mistake similar to the following. INCORRECT

21. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Subtraction with Rational Expressions Notes (cont.) By using parentheses, you can avoid such mistakes. CORRECT

22. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Subtracting Rational Expressions with Different Denominators Find each difference and reduce if possible. Assume that no denominator has a value of 0. Solution

23. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Subtracting Rational Expressions with Different Denominators (cont.)

24. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Subtracting Rational Expressions with Different Denominators (cont.) Solution

25. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Subtracting Rational Expressions with Different Denominators (cont.)

26. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Subtracting Rational Expressions with Different Denominators (cont.) Solution

27. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Subtracting Rational Expressions with Different Denominators (cont.) Now subtract these two expressions with LCD = (x + 5) (x + 4).

28. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Subtracting Rational Expressions with Different Denominators (cont.) Solution

29. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Subtracting Rational Expressions with Different Denominators (cont.)

30. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problems Perform the indicated operations and reduce if possible. Assume that no denominator is 0. 1. 2. 3. 4.

31. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problem Answers 1.2. 3. 4.